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Compression and microbunching of electron beams by ultra-intense laser pulses
Physics Letters A 353 (2006) 505–511 www.elsevier.com/locate/pla
Compression and microbunching of electron beams by ultra-intense laser pulses Victor V. Kulagin a,1 , Vladimir A. Cherepenin b , Min Sup Hur a , Hyyong Suk a,∗ a Center for Advanced Accelerators, KERI, Changwon 641-120, Republic of Korea b Institute of Radioengineering and Electronics RAS, Mohovaya 11, Moscow 125009, Russia
Received 13 October 2005; received in revised form 21 December 2005; accepted 7 January 2006 Available online 18 January 2006 Communicated by F. Porcelli
Abstract The formation of coherent structures, induced by a super-intense plane electromagnetic wave with a sharp rising edge in an ensemble of electrons (electron beam) in vacuum, is considered. The theory describing this process is elaborated. It is shown that the laser pulse can strongly compress the electron beam and also generate fast density modulations (microbunching) in it. Depending on the duration of a laser pulse front, two harmonics can be present simultaneously in longitudinal density modulations of the electron beam—one with laser wavelength and the other with half of the laser wavelength. By changing the form of the laser pulse envelope, one can control the average density of the electron beam (slow density modulation). By varying the laser pulse amplitude and initial length of the electron beam, it is possible to change the number of microbunches in the compressed electron beam, and for certain conditions only one electron bunch can be produced with ultrashort length smaller than the laser wavelength (attosecond length electron beam). The results of the theory are compared with 1D PIC (particle-in-cell) simulations and a good agreement is found. © 2006 Elsevier B.V. All rights reserved. PACS: 41.75.Jv; 41.75.Ht; 41.85.-p Keywords: Laser acceleration of electrons; Ultrashort electron beams; Electron microbunches; Super-intense laser pulse
1. Introduction Interaction of electrons with intense electromagnetic waves in vacuum, being a textbook problem [1], remains an issue of the day in current research. Main directions here are an acceleration of the electrons by an external electromagnetic wave [2–15] and generation of radiation by relativistic electrons [16–18]. To accelerate an electron by a laser pulse in vacuum, it is necessary to use a strong temporal or spatial inhomogeneity of the field [2–4,6,8–11,13]. A plain wave with * Corresponding author.
E-mail addresses: [email protected] (V.V. Kulagin), [email protected] (V.A. Cherepenin), [email protected] (M.S. Hur), [email protected] (H. Suk). 1 On leave from Sternberg Astronomical Institute of Moscow State University, Russia. 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.01.011
a smooth time profile cannot accelerate the electron [1], in the sense that after interaction with such a pulse the electron energy remains unchanged (however, during interaction an electron can be highly relativistic). Another direction of research is acceleration of the electrons by an electromagnetic wave in the presence of accelerating structures, which provide external magnetic or electric fields [the inverse free-electron laser (IFEL) mechanism of acceleration] [19–22]. An interesting feature of the IFEL mechanism is microbunching of accelerated electron beam, i.e. the electron density of the beam in the output of accelerator becomes modulated longitudinally with the period of the laser wavelength [19–25]. This feature is also typical for free-electron lasers and provides a very efficient generation of electromagnetic radiation there [16–18,26,27]. Alternatively, the microbunching can be realized by using two-frequency lasers [28]. Moreover, in laser-plasma experiments with foils, microbunched electron beams can also be
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produced [29,30]. However, the modulation period here is equal to the half of the laser wavelength because in this case the microbunching is generated by the nonlinear component v × B of the Lorentz force in the laser pulse. The goal of this Letter is to investigate the dynamics and characteristics of coherent structures induced by a super-intense plane electromagnetic wave with a sharp rising edge in an ensemble of electrons (electron beam) in vacuum. From an analytical theory and one-dimensional (1D) particle-in-cell (PIC) simulations, we will show that the action of a super-intense plane electromagnetic wave on the electrons consists not only in the ordinary acceleration during interaction [1]. Other interesting features are a strong longitudinal compression of the electron beam and a generation of density modulations (microbunching) in the electron beam. Moreover, depending on the duration of a laser pulse front, two spatial harmonics can appear simultaneously in longitudinal density modulations of the electron beam—one with laser wavelength and another with half of the laser wavelength. Besides, if the initial electron beam is too short or the laser pulse is too strong, only one electron bunch can be produced with ultrashort length smaller than the laser wavelength. It is necessary to note that investigation for the mechanism of bunching of electrons in a super-intense electromagnetic wave has an independent interest since such a mechanism can be very important in astrophysics where field amplitudes are extremely large. Besides, it can be used in a laboratory for generation of high-frequency electromagnetic radiation [31–33] (and in this case, extraction of the electrons from the electromagnetic wave is not necessary) or for production of relativistic microbunched electron beams or ultrashort single electron bunches. In latter cases, relativistic electrons can be extracted from the electromagnetic wave just at their energy peak with the help of a plasma separator [34–36]. Now table-top terawatt lasers [37] are widely used in experiments, and the perspectives for power increase in future installations are even more exciting [38]. In such fields, electrons can acquire energy from hundreds of MeV to several GeV and can be used in many areas of physics, technology, medicine, etc. In this Letter we consider electron beams with not very high electron density such that Coulomb forces between the electrons and all collective effects can be neglected. Besides, the radiation of the electrons are also omitted. So the problem is mathematically equivalent to the motion of a single electron in a plane electromagnetic wave. As was stated above, however, our prime interest is in considering the mutual motion of the whole electron ensemble in the electromagnetic field. The Letter is organized as follows: equations of motion for the electrons and their solutions are presented in Section 2. In Section 3 the physical basis for longitudinal compression of the electron beam is discussed and the expression for density modulations is derived. In Section 4 results for generation of ultrashort electron bunches are specified and comparison between the theory and 1D PIC simulations is presented. The same data are also presented for microbunched electron beams. Section 5 concludes the discussion.
2. Motion of a single electron in an electromagnetic wave Let a linearly y-polarized plane electromagnetic wave with frequency ω and wave vector k (k = ω/c) be incident along the z axis. Then the field has the form Ey (ωt − kz) = E0 ey (ωt − kz), where E0 is the amplitude of the wave and the function ey describes the wave envelope. The equations of motion for the electrons have the following form dpy /dω t = −a0 ey (1 − βz ) sin(θ + θ0 ), dpz /dω t = −a0 βy ey sin(θ + θ0 ), dγ /dω t = −a0 βy ey sin(θ + θ0 ),
(1)
where θ + θ0 = ωt − kz(t) + θ0 is the phase of the field at the electron position, θ0 is the initial phase of the field, and a0 = |e|E0 /(mωc) (e and m are the charge and the mass of an electron, c is the speed of light). Here, the normalized momenta are given by py,z = γβy,z = [1 − (v/c)2 ]−1/2 vy,z /c, where v is the velocity of an electron. These equations are valid for all electrons, and the only difference is the phase θ of the laser field due to different initial positions of the electrons. Eqs. (1) can be easily solved analytically [1] giving the invariant κ = γ −pz = γ (1−βz ) = 1 for the zero initial momenta of the electrons. Besides, taking the variable θ as an independent variable, one has for θ 0 (we suppose here and below that the electrons are at rest initially) θ py (θ ) = −a0
ey (θ ) sin(θ + θ0 ) dθ ,
0
pz (θ ) =
py2 2
γ (θ ) = 1 +
, py2 2
, θ
ky(θ) = ky0 +
py (θ ) dθ ,
0
1 kz(θ ) = kz0 + 2
θ
py2 (θ ) dθ = kz0 + kZ(θ ),
0
ωt = θ + kz(θ ).
(2)
Therefore, all the parameters of motion are defined by py (θ ), and θ in its turn can be defined as a function of ωt and kz0 (Lagrangian description). In a special case of the given field, considered in this Letter, the variable θ depends only on combination ωt − kz0 [cf. two last equations in the system (2)] that just accounts for the time necessary for the laser pulse to reach an electron with coordinate kz0 . Moreover, the longitudinal coordinates of all electrons are defined just by one function θ Z(θ ) = 12 0 py2 (θ ) dθ , and there are no intersections of the trajectories. So the laser pulse picks up all electrons. From Eqs. (2) one can conclude that the solutions essentially depend on the form of the envelope ey (t). Below, to elucidate the role of a laser pulse front, we will consider laser pulses with a sharp front and a long body with constant amplitude. To be
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concrete let us consider at first the linearly growing pulse front: ⎧ θ < 0, ⎨ 0, ey (θ ) = θ/τf , 0 θ τf , (3) ⎩ 1, θ > τf . We are interested in the motion of an electron at the top of the laser pulse, i.e., when the front already outstrips the electron. Then one has for θ τf sin(τf /2) py (θ ) = −a0 cos(τf /2 + θ0 ) − cos(θ + θ0 ) . (4) τf /2 The first term in the brackets in the right-hand side of Eq. (4) is constant. Its absolute value cannot be larger than one. Besides, it is vanishing for τf 2π and depends strongly on the initial phase of the field θ0 . So the transverse momentum of an electron, py , can have a nonzero constant component in the field of an ultrashort laser pulse. Then, the longitudinal momentum pz and the relativistic factor γ will consist, according to Eqs. (2), of terms, varying harmonically with θ with single and double frequencies. For long laser pulse front τf 2π (adiabatic pulse), only the double frequency term can survive, while for ultrashort pulse front τf < 2π (nonadiabatic pulse) the single frequency term is also present. Similar conclusions can be made for the other forms of the pulse envelope. It is necessary to note that laser pulses with very sharp fronts τf 2π have been already generated in experiment [39,40]. Moreover, the phase θ0 , which is called the carrier-envelope phase, can be easily controlled for ultrashort pulses [41]. So in principle nonzero constant term in py can be realized experimentally. It is necessary to mention that the same result (nonzero constant term in py ) can be achieved even for an adiabatic laser pulse by using initially relativistic electron beam with py0 = 0. However, in this case preliminary acceleration of the electrons is essential. For τf → 0 and θ0 = 0 (a step-like envelope) one has [6,32, 42] from Eqs. (2) and (4) py (θ ) = a0 (cos θ − 1), a02 (cos θ − 1)2 , 2 a 2 (cos θ − 1)2 , γ (θ ) = 1 + 0 2 ky = ky0 + a0 (sin θ − θ ),
sin 2θ 2 3θ kz = kz0 + a0 − sin θ + . 4 8
pz (θ ) =
(5)
These solutions can be used as benchmarks for estimating scales of different variables. The longitudinal displacement of an electron during laser period (θ changes by 2π ) is equal for this case to L2π = 3a02 λ/4,
(6)
and for a0 1 it can be considerably larger than the laser wavelength. Period of electron motion in the laboratory frame is T2π = (1 + 3a02 /4)λ/c and can be considerably larger than the laser period (up to hundreds of femtoseconds). Maximum
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values for the transverse py and the longitudinal pz momenta and for γ are achieved for θ = π(1 + 2n), n = 0, 1, 2, . . . , and equal to 2a0 , 2a02 and 1 + 2a02 correspondingly. So maximal electron energy in the field with a0 = 5 is 25.5 MeV and with a0 = 10 it is 100.5 MeV that can be enough for many applications. Same estimates for adiabatic laser pulse give the energies 6.75 MeV and 25.5 MeV that are not so bad. The averaged (over laser period) longitudinal velocity of the electron is (βz )av = 1 − (1 + 3a02 /4)−1 for the step-like envelope and can be close to 1. It is necessary to mention that with account for the reaction of the self-radiation of the electrons, the constant term in Eq. (4) decreases, and the rate of decrease is greater for larger density of electrons [33,42,43]. Therefore, the presence of a constant term in py is difficult to observe in a dense media, besides, very fast measurements are necessary. 3. Compression and microbunching of the electron layer Let us now consider why a laser pulse can compress the electron beam longitudinally [32,42]. Let one electron has the coordinate z01 and the other has the coordinate z02 , and l0 = z02 − z01 is considerably smaller than the accelerating laser wavelength λ. Then, z1 (t) = z01 + Z(t − z01 /c) and z2 (t) = z02 + Z(t − z02 /c) are the evolutions of the electrons’ coordinates in time [cf. Eq. (2)], and t = (z02 − z01 )/c = l0 /c is the delay required for the laser pulse to propagate from the point z01 to the point z02 . For l0 λ, the delay t is considerably smaller than the laser period, and the distance between two electrons is l(t) = z2 − z1 = l0 + Z(t − z02 /c) − Z(t − z01 /c) l0 − (dZ/dt) t = l0 1 − βz (t) = l0 /γ (t)
(7)
because of the invariant value of κ. This equality holds for an arbitrary laser pulse envelope, as long as the motion of the electrons can be considered as being that in the given field. It is necessary to note that the compression by γ is just that required according to the special theory of relativity for the length of a body which is initially at rest and then is accelerated to some relativistic velocity [1]. Thus, the distance between two electrons will oscillate according to Eqs. (2) and (5), and the smallest value for a step-like envelope lmin l0 / 2a02 (8) will be achieved for the largest values of γ . The value of lmin can be considerably smaller than the initial distance l0 . Local density n(t) = n0 l0 / l(t) = n0 γ (t), corresponding to initial points z01 and z02 , increases in its turn and can be considerably larger than the initial density n0 . The averaged density of a compressed electron beam can be easily estimated from Eqs. (2). Actually, two electrons separated initially by z02 − z01 = L2π (corresponding to the change of θ by 2π ) after compression occurs at a distance λ one from another, so, e.g., for the step-like envelope nav = n0 (1 + 3a02 /4) and can be high enough. Adiabatic laser pulse with a smooth envelope can provide nav = n0 (1 + a02 /4). So the electron beam is strongly compressed by
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a laser pulse in the longitudinal direction, and this compression takes place just at the front of the laser pulse. This result explains the so-called “snowplow” effect obtained in the PIC simulation [44] for the exotic unipolar accelerating laser pulse and for the accelerating sharp-rise envelope laser pulse [13,33, 45]. Let us now formally derive the expression for the electron density of the beam at time t as a function of the longitudinal coordinate z. Taking a conservation of charge into account, one has
d[kz(z0 , t)] −1 , n(z0 , t) = n0 (z0 ) (9) d(kz0 ) where n0 (z0 ) is the initial density of the beam at point z0 . In our case, the longitudinal coordinate z is a function of θ so one has from Eqs. (2) d[kz(z0 , t)] ∂[kZ(θ)] ∂(ωt) −1 =1− · d(kz0 ) ∂θ ∂θ −1 pz (θ ) =1− (10) = γ (θ ) 1 + pz (θ ) and now θ is the function of t and kz0 . Then from Eqs. (9) and (10) one has n(z0 , t) = n0 (z0 )γ (θ ),
(11)
where θ is defined by the last equation of Eqs. (2). To derive the density distribution at time t, depending on coordinates z, one needs to switch from parameter z0 to the independent parameter z {lying, of course, inside the allowed interval [zmin (t), zmax (t)]}. Then one has n(z, t) = n0 z0 (z) γ (ωt − kz), z ∈ zmin (t), zmax (t) . (12) This expression has the simplest form when the initial density is constant inside some interval z0 ∈ [0, zmax ] and zero outside. Then the density distribution is n(z, t)/n0 = γ (ωt − kz), z ∈ zmin (t), zmax (t) , (13) and the ends of the interval are just the coordinates of the left and the right electrons at time t [and can be derived from Eqs. (2) along with the function γ ]. From Eq. (13) it is obvious that, in the electron beam, there is a density wave, which is running inside the laser pulse with the same speed. Of course, each electron in this density wave has the speed less than c, and slips along the density ripple alternately entering the regions with small and large density. The energy and longitudinal momentum pz = γ − 1 of an electron are maximal where the density is high and minimal in the regions with small density. The form of the density modulation is defined solely by the transverse momentum py [cf. Eq. (2)]. So if py is without a constant term, then the period of density modulation is λ/2. Otherwise there will be two spatial harmonics with periods λ/2 and λ, and relative amplitudes of these harmonics will be defined by the parameters of the laser pulse front. Besides, by changing the form of the laser pulse envelope, one can control the average density of the electron beam (produce slow density modulation). For a constant amplitude of the
laser pulse, the average density will be constant (the Coulomb forces will give a slow density modulation of the beam even in the constant amplitude field, however, in our analysis, this effect is omitted because a small enough electron density of the beam is considered). In passing, it is interesting to note that the measurement of the amplitudes of density modulation harmonics by, e.g., detecting the coherent transition radiation of the beam [23,26,27] can give very valuable information about the parameters of the laser pulse front. 4. Density distributions for electron beams: Theory versus PIC simulations Let us at first consider the formation of a single short electron bunch with the length smaller than the laser wavelength. To realize this, the initial length Lin of the electron beam must be smaller than the maximal length L2π defined for the laser pulse. Then, the density distribution will change its form with the slow time scale T2π . For Lin λ, the compressed length of the beam will oscillate between Lin and Lin /γ and the density will oscillate between n0 and n0 γ . For Lin ∼ L2π , the density distribution will have the width about λ, and will have one or two maxima. It is necessary to note that to output the electron beam from the laser pulse, one needs to place a plasma separator with an accuracy much better than L2π (that seems not to be a very difficult problem). Below, the electron density distributions inside the laser pulse with the amplitude a0 = 5 are presented for different times. In Fig. 1, the laser pulse has the step-like envelope; in Fig. 2, the laser front has a Gaussian shape and the duration 5λ (at half maximum). The values L2π for both cases can be defined from Eqs. (2) and (6). These values are about 19λ and 6λ. In Fig. 1, the initial length of the beam is 0.2λ, and it is 3λ in Fig. 2. In the same figures, the results of the PIC simulations with the one-dimensional version of XOOPIC code [46] are also presented (solid cyan lines). The densities of the electron beam were n0 = 1015 cm−3 for Fig. 1 and n0 = 1012 cm−3 for Fig. 2, initial velocity of the electrons is equal to zero. There were 8100 (Fig. 1) or 900 (Fig. 2) spatial grid points on the laser wavelength λ, and there were 10 (Fig. 1) or 20 (Fig. 2) macroparticles in one grid cell. From these figures, one can conclude that the theory gives a very good agreement with the PIC simulations. The excess noise in Fig. 1(d) and (e) can be explained by not high enough accuracy of the PIC simulations (to remove the noise it is necessary to increase the number of spatial grid points on the laser wavelength). From Fig. 1, one can conclude that the electron beam is almost monoenergetic at the point of maximum compression [Fig. 1(c)]. More exactly, the spread in the energy is smaller for the smaller initial electron beam length and the larger amplitude of the laser pulse. Besides, at the points of intermediate compression, there is a modulation in the energy of the electrons [cf. Fig. 1(b) and (d), the energy is equal to the density according to Eq. (13)]. This energy modulation allows, in principle, to realize a further compression with the help of a magnetic chi-
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Fig. 1. Pulsation of the short electron beam in the field of the ultra-intense laser pulse with a step-like envelope and a0 = 5 (blue dots—theory, solid cyan lines—PIC simulations). Electron density (in units of n0 ) is presented for different times: ωt 1.26, 6.28, 62.83, 116.24, 124.41 for (a)–(e). Distance on the horizontal axis is measured from the head of the laser pulse, which is located in kz = 0. Note different vertical scales for plots (a)–(e). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
cane. In Fig. 2, the gradual compression of the electron beam by the laser front can be easily traced. Here also, the average density of the electron beam is modulated [Fig. 2(b) and (c)]. Formation of the microbunched electron beams is presented in Fig. 3. In Fig. 3(a), the envelope of the laser pulse is a steplike; in Fig. 3(b), the duration of the laser pulse front is 0.89λ (at half maximum), and in Fig. 3(c) it is 2.2λ. The initial length Lin of the electron beam is equal to 100λ (0.1 mm for λ = 1 µ). The results of the PIC simulations are presented for the case of Fig. 3(c) (solid cyan line). The electron density for this case is n0 = 1010 cm−3 . There are 510 spatial grid points on the laser wavelength λ, and there are 20 macroparticles in one grid cell. Again, the results of the theory and the PIC simulation are almost the same. From these figures, the microbunching of the electron beams is evident. Besides, for the longitudinal modulation of the electron density, the relative amplitudes of two spatial harmonics with periods λ and λ/2 (frequencies ω and 2ω) depend on the duration of the laser pulse front as predicted by the theory.
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Fig. 2. Compression of the electron beam by the laser pulse front with duration 5λ (at half maximum) and a0 = 5 (blue dots—theory, solid cyan lines—PIC simulations). The laser field at ωt = 188.5 is presented in (a); (b)–(g)—electron density distributions (in units of n0 ) for different times ωt 119.41, 128.84, 150.83, 216.8, 270.21, 282.78. Note different vertical scales for plots (b)–(g). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Again, one can see that by changing the laser pulse envelope, it is possible to control the average density of the electron beam [cf. slow density modulation at the head of the electron beam in Fig. 3(c)]. 5. Discussions of results and conclusions It was demonstrated above that the ultra-intense laser pulses can generate microbunched relativistic electron beams or ultrashort electron bunches. Let us discuss now the conditions, which are necessary for practical realization of this method, and possible parameters of resulting electron beams. The required laser power is determined by the amplitude a0 and the diameter of the laser beam d, which has to be at least not smaller than the
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Fig. 3. Microbunching of the electron beam by laser pulses with different front duration: (a)—the step-like envelope, (b) and (c)—the duration of the laser pulse front at half maximum is 0.89λ and 2.2λ. The electron densities (in units of n0 ) are presented for ωt 100 · 2π . Note different vertical scales in plots (a)–(c).
transverse displacement y of the electrons in the beam. Therefore from Eqs. (2), one can conclude that the required value of d depends on the duration of the laser pulse front and also on the initial length of the electron beam (through maximal value of θ ). One can estimate from Eqs. (5) that for the generation of the ultrashort electron bunch by the laser pulse with a step-like envelope, it is necessary to have d a0 , where d is measured in λ, and θ = 2π is used. Then P = (da0 )2 P1 = a04 P1 , where P1 is the laser power required to create a0 = 1 in the beam spot with diameter λ (about 14 GW for λ 1 µ). So, for a 1 PW laser pulse, which is available now, the maximum value of a0 can be about 17 and d 17 µ, giving the maximal electron energy on the order of 300 MeV. If the laser power is 1 EW (1018 W) [38] then a0 100 (d 0.1 mm) and the electron energy is 10 GeV. For the generation of a microbunched electron beam by a laser pulse with a step-like envelope, the requirements on d are more strict, namely d must be proportional to the maximal value of θ [cf. Eqs. (5)]. Thus, for the generation of the beam with, e.g., 5 humps one needs to have θmax = 5 · 2π , and d 5a0 . So the maximal value of a0 for the petawatt laser is only a0 8 (d 40 µ). For laser pulses with a smooth front,
the constant term in the transverse momentum is vanishing, so the transverse coordinate of an electron is bounded, and the required value of d is independent on the length of the beam [cf. Eqs. (2)]. So all estimates for θ = 2π (more exactly, for θ = 1) and a step-like envelope are valid also for this case. The longitudinal scale of the method is defined by the value of L2π . Again, for the ultrashort electron bunch generated by the laser pulse with a step-like envelope, L2π is about 225 µ for 1 PW laser pulse and 7.5 mm for 1 EW laser pulse. If the required number of humps in the beam is greater than 1 then this length is increased proportionally. It is necessary to note that the generalization of the above results on the case of nonzero initial velocity of the electrons is straightforward. Also, all the results can be easily expanded on the case of short laser pulses. In this case, noticeable increase of the electron density will be only in the regions where the amplitude of the laser pulse is large enough. For this region, all the results for microbunching are applicable. When the laser pulse will go out of the electron beam, the electron density will return to its initial value. In conclusion, the formation of the coherent structures, induced by a super-intense plane electromagnetic wave with a sharp rising edge in an ensemble of electrons (electron beam) in vacuum, was considered. The theory describing this process was elaborated. It was shown that the laser pulse can strongly compress the electron beam, and this process takes place just at the front of the laser pulse. If the laser pulse has a slowly varying envelope then the density of the compressed electron beam has a slow modulation, because the degree of compression and the average density of the electron beam depend on the laser pulse amplitude. Also, an ultraintense laser pulse generates fast density modulation (microbunching) in the electron beam. Depending on the duration of a laser pulse front, two spatial harmonics can be present simultaneously in the longitudinal density modulations of the electron beam—one with laser wavelength and the other with half of the laser wavelength. Besides, by varying the laser pulse amplitude and the initial length of the electron beam, it is possible to change the number of microbunches in the compressed electron beam, so that for certain conditions only one electron microbunch can be produced with an ultrashort length on the attosecond level. For some moments, the energy of the electrons in this single bunch is strongly modulated, allowing further compression of the bunch with a magnetic chicane. The results of the theory were compared with the 1D PIC simulations and a good agreement was found. Acknowledgements This work was partially supported by the CRI Program of the Korean Ministry of Science and Technology, by the INTAS project 01-0233 (V.A.C.), and by RFBR project 05-02-17297-a. References [1] L.D. Landau, E.M. Lifshitz, Teoriya Polya, Nauka, Moscow, 1988, The Classical Theory of Fields, Pergamon, Oxford, 1975.
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Compression and microbunching of electron beams by ultra-intense laser pulses Victor V. Kulagin a,1 , Vladimir A. Cherepenin b , Min Sup Hur a , Hyyong Suk a,∗ a Center for Advanced Accelerators, KERI, Changwon 641-120, Republic of Korea b Institute of Radioengineering and Electronics RAS, Mohovaya 11, Moscow 125009, Russia
Received 13 October 2005; received in revised form 21 December 2005; accepted 7 January 2006 Available online 18 January 2006 Communicated by F. Porcelli
Abstract The formation of coherent structures, induced by a super-intense plane electromagnetic wave with a sharp rising edge in an ensemble of electrons (electron beam) in vacuum, is considered. The theory describing this process is elaborated. It is shown that the laser pulse can strongly compress the electron beam and also generate fast density modulations (microbunching) in it. Depending on the duration of a laser pulse front, two harmonics can be present simultaneously in longitudinal density modulations of the electron beam—one with laser wavelength and the other with half of the laser wavelength. By changing the form of the laser pulse envelope, one can control the average density of the electron beam (slow density modulation). By varying the laser pulse amplitude and initial length of the electron beam, it is possible to change the number of microbunches in the compressed electron beam, and for certain conditions only one electron bunch can be produced with ultrashort length smaller than the laser wavelength (attosecond length electron beam). The results of the theory are compared with 1D PIC (particle-in-cell) simulations and a good agreement is found. © 2006 Elsevier B.V. All rights reserved. PACS: 41.75.Jv; 41.75.Ht; 41.85.-p Keywords: Laser acceleration of electrons; Ultrashort electron beams; Electron microbunches; Super-intense laser pulse
1. Introduction Interaction of electrons with intense electromagnetic waves in vacuum, being a textbook problem [1], remains an issue of the day in current research. Main directions here are an acceleration of the electrons by an external electromagnetic wave [2–15] and generation of radiation by relativistic electrons [16–18]. To accelerate an electron by a laser pulse in vacuum, it is necessary to use a strong temporal or spatial inhomogeneity of the field [2–4,6,8–11,13]. A plain wave with * Corresponding author.
E-mail addresses: [email protected] (V.V. Kulagin), [email protected] (V.A. Cherepenin), [email protected] (M.S. Hur), [email protected] (H. Suk). 1 On leave from Sternberg Astronomical Institute of Moscow State University, Russia. 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.01.011
a smooth time profile cannot accelerate the electron [1], in the sense that after interaction with such a pulse the electron energy remains unchanged (however, during interaction an electron can be highly relativistic). Another direction of research is acceleration of the electrons by an electromagnetic wave in the presence of accelerating structures, which provide external magnetic or electric fields [the inverse free-electron laser (IFEL) mechanism of acceleration] [19–22]. An interesting feature of the IFEL mechanism is microbunching of accelerated electron beam, i.e. the electron density of the beam in the output of accelerator becomes modulated longitudinally with the period of the laser wavelength [19–25]. This feature is also typical for free-electron lasers and provides a very efficient generation of electromagnetic radiation there [16–18,26,27]. Alternatively, the microbunching can be realized by using two-frequency lasers [28]. Moreover, in laser-plasma experiments with foils, microbunched electron beams can also be
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produced [29,30]. However, the modulation period here is equal to the half of the laser wavelength because in this case the microbunching is generated by the nonlinear component v × B of the Lorentz force in the laser pulse. The goal of this Letter is to investigate the dynamics and characteristics of coherent structures induced by a super-intense plane electromagnetic wave with a sharp rising edge in an ensemble of electrons (electron beam) in vacuum. From an analytical theory and one-dimensional (1D) particle-in-cell (PIC) simulations, we will show that the action of a super-intense plane electromagnetic wave on the electrons consists not only in the ordinary acceleration during interaction [1]. Other interesting features are a strong longitudinal compression of the electron beam and a generation of density modulations (microbunching) in the electron beam. Moreover, depending on the duration of a laser pulse front, two spatial harmonics can appear simultaneously in longitudinal density modulations of the electron beam—one with laser wavelength and another with half of the laser wavelength. Besides, if the initial electron beam is too short or the laser pulse is too strong, only one electron bunch can be produced with ultrashort length smaller than the laser wavelength. It is necessary to note that investigation for the mechanism of bunching of electrons in a super-intense electromagnetic wave has an independent interest since such a mechanism can be very important in astrophysics where field amplitudes are extremely large. Besides, it can be used in a laboratory for generation of high-frequency electromagnetic radiation [31–33] (and in this case, extraction of the electrons from the electromagnetic wave is not necessary) or for production of relativistic microbunched electron beams or ultrashort single electron bunches. In latter cases, relativistic electrons can be extracted from the electromagnetic wave just at their energy peak with the help of a plasma separator [34–36]. Now table-top terawatt lasers [37] are widely used in experiments, and the perspectives for power increase in future installations are even more exciting [38]. In such fields, electrons can acquire energy from hundreds of MeV to several GeV and can be used in many areas of physics, technology, medicine, etc. In this Letter we consider electron beams with not very high electron density such that Coulomb forces between the electrons and all collective effects can be neglected. Besides, the radiation of the electrons are also omitted. So the problem is mathematically equivalent to the motion of a single electron in a plane electromagnetic wave. As was stated above, however, our prime interest is in considering the mutual motion of the whole electron ensemble in the electromagnetic field. The Letter is organized as follows: equations of motion for the electrons and their solutions are presented in Section 2. In Section 3 the physical basis for longitudinal compression of the electron beam is discussed and the expression for density modulations is derived. In Section 4 results for generation of ultrashort electron bunches are specified and comparison between the theory and 1D PIC simulations is presented. The same data are also presented for microbunched electron beams. Section 5 concludes the discussion.
2. Motion of a single electron in an electromagnetic wave Let a linearly y-polarized plane electromagnetic wave with frequency ω and wave vector k (k = ω/c) be incident along the z axis. Then the field has the form Ey (ωt − kz) = E0 ey (ωt − kz), where E0 is the amplitude of the wave and the function ey describes the wave envelope. The equations of motion for the electrons have the following form dpy /dω t = −a0 ey (1 − βz ) sin(θ + θ0 ), dpz /dω t = −a0 βy ey sin(θ + θ0 ), dγ /dω t = −a0 βy ey sin(θ + θ0 ),
(1)
where θ + θ0 = ωt − kz(t) + θ0 is the phase of the field at the electron position, θ0 is the initial phase of the field, and a0 = |e|E0 /(mωc) (e and m are the charge and the mass of an electron, c is the speed of light). Here, the normalized momenta are given by py,z = γβy,z = [1 − (v/c)2 ]−1/2 vy,z /c, where v is the velocity of an electron. These equations are valid for all electrons, and the only difference is the phase θ of the laser field due to different initial positions of the electrons. Eqs. (1) can be easily solved analytically [1] giving the invariant κ = γ −pz = γ (1−βz ) = 1 for the zero initial momenta of the electrons. Besides, taking the variable θ as an independent variable, one has for θ 0 (we suppose here and below that the electrons are at rest initially) θ py (θ ) = −a0
ey (θ ) sin(θ + θ0 ) dθ ,
0
pz (θ ) =
py2 2
γ (θ ) = 1 +
, py2 2
, θ
ky(θ) = ky0 +
py (θ ) dθ ,
0
1 kz(θ ) = kz0 + 2
θ
py2 (θ ) dθ = kz0 + kZ(θ ),
0
ωt = θ + kz(θ ).
(2)
Therefore, all the parameters of motion are defined by py (θ ), and θ in its turn can be defined as a function of ωt and kz0 (Lagrangian description). In a special case of the given field, considered in this Letter, the variable θ depends only on combination ωt − kz0 [cf. two last equations in the system (2)] that just accounts for the time necessary for the laser pulse to reach an electron with coordinate kz0 . Moreover, the longitudinal coordinates of all electrons are defined just by one function θ Z(θ ) = 12 0 py2 (θ ) dθ , and there are no intersections of the trajectories. So the laser pulse picks up all electrons. From Eqs. (2) one can conclude that the solutions essentially depend on the form of the envelope ey (t). Below, to elucidate the role of a laser pulse front, we will consider laser pulses with a sharp front and a long body with constant amplitude. To be
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concrete let us consider at first the linearly growing pulse front: ⎧ θ < 0, ⎨ 0, ey (θ ) = θ/τf , 0 θ τf , (3) ⎩ 1, θ > τf . We are interested in the motion of an electron at the top of the laser pulse, i.e., when the front already outstrips the electron. Then one has for θ τf sin(τf /2) py (θ ) = −a0 cos(τf /2 + θ0 ) − cos(θ + θ0 ) . (4) τf /2 The first term in the brackets in the right-hand side of Eq. (4) is constant. Its absolute value cannot be larger than one. Besides, it is vanishing for τf 2π and depends strongly on the initial phase of the field θ0 . So the transverse momentum of an electron, py , can have a nonzero constant component in the field of an ultrashort laser pulse. Then, the longitudinal momentum pz and the relativistic factor γ will consist, according to Eqs. (2), of terms, varying harmonically with θ with single and double frequencies. For long laser pulse front τf 2π (adiabatic pulse), only the double frequency term can survive, while for ultrashort pulse front τf < 2π (nonadiabatic pulse) the single frequency term is also present. Similar conclusions can be made for the other forms of the pulse envelope. It is necessary to note that laser pulses with very sharp fronts τf 2π have been already generated in experiment [39,40]. Moreover, the phase θ0 , which is called the carrier-envelope phase, can be easily controlled for ultrashort pulses [41]. So in principle nonzero constant term in py can be realized experimentally. It is necessary to mention that the same result (nonzero constant term in py ) can be achieved even for an adiabatic laser pulse by using initially relativistic electron beam with py0 = 0. However, in this case preliminary acceleration of the electrons is essential. For τf → 0 and θ0 = 0 (a step-like envelope) one has [6,32, 42] from Eqs. (2) and (4) py (θ ) = a0 (cos θ − 1), a02 (cos θ − 1)2 , 2 a 2 (cos θ − 1)2 , γ (θ ) = 1 + 0 2 ky = ky0 + a0 (sin θ − θ ),
sin 2θ 2 3θ kz = kz0 + a0 − sin θ + . 4 8
pz (θ ) =
(5)
These solutions can be used as benchmarks for estimating scales of different variables. The longitudinal displacement of an electron during laser period (θ changes by 2π ) is equal for this case to L2π = 3a02 λ/4,
(6)
and for a0 1 it can be considerably larger than the laser wavelength. Period of electron motion in the laboratory frame is T2π = (1 + 3a02 /4)λ/c and can be considerably larger than the laser period (up to hundreds of femtoseconds). Maximum
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values for the transverse py and the longitudinal pz momenta and for γ are achieved for θ = π(1 + 2n), n = 0, 1, 2, . . . , and equal to 2a0 , 2a02 and 1 + 2a02 correspondingly. So maximal electron energy in the field with a0 = 5 is 25.5 MeV and with a0 = 10 it is 100.5 MeV that can be enough for many applications. Same estimates for adiabatic laser pulse give the energies 6.75 MeV and 25.5 MeV that are not so bad. The averaged (over laser period) longitudinal velocity of the electron is (βz )av = 1 − (1 + 3a02 /4)−1 for the step-like envelope and can be close to 1. It is necessary to mention that with account for the reaction of the self-radiation of the electrons, the constant term in Eq. (4) decreases, and the rate of decrease is greater for larger density of electrons [33,42,43]. Therefore, the presence of a constant term in py is difficult to observe in a dense media, besides, very fast measurements are necessary. 3. Compression and microbunching of the electron layer Let us now consider why a laser pulse can compress the electron beam longitudinally [32,42]. Let one electron has the coordinate z01 and the other has the coordinate z02 , and l0 = z02 − z01 is considerably smaller than the accelerating laser wavelength λ. Then, z1 (t) = z01 + Z(t − z01 /c) and z2 (t) = z02 + Z(t − z02 /c) are the evolutions of the electrons’ coordinates in time [cf. Eq. (2)], and t = (z02 − z01 )/c = l0 /c is the delay required for the laser pulse to propagate from the point z01 to the point z02 . For l0 λ, the delay t is considerably smaller than the laser period, and the distance between two electrons is l(t) = z2 − z1 = l0 + Z(t − z02 /c) − Z(t − z01 /c) l0 − (dZ/dt) t = l0 1 − βz (t) = l0 /γ (t)
(7)
because of the invariant value of κ. This equality holds for an arbitrary laser pulse envelope, as long as the motion of the electrons can be considered as being that in the given field. It is necessary to note that the compression by γ is just that required according to the special theory of relativity for the length of a body which is initially at rest and then is accelerated to some relativistic velocity [1]. Thus, the distance between two electrons will oscillate according to Eqs. (2) and (5), and the smallest value for a step-like envelope lmin l0 / 2a02 (8) will be achieved for the largest values of γ . The value of lmin can be considerably smaller than the initial distance l0 . Local density n(t) = n0 l0 / l(t) = n0 γ (t), corresponding to initial points z01 and z02 , increases in its turn and can be considerably larger than the initial density n0 . The averaged density of a compressed electron beam can be easily estimated from Eqs. (2). Actually, two electrons separated initially by z02 − z01 = L2π (corresponding to the change of θ by 2π ) after compression occurs at a distance λ one from another, so, e.g., for the step-like envelope nav = n0 (1 + 3a02 /4) and can be high enough. Adiabatic laser pulse with a smooth envelope can provide nav = n0 (1 + a02 /4). So the electron beam is strongly compressed by
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a laser pulse in the longitudinal direction, and this compression takes place just at the front of the laser pulse. This result explains the so-called “snowplow” effect obtained in the PIC simulation [44] for the exotic unipolar accelerating laser pulse and for the accelerating sharp-rise envelope laser pulse [13,33, 45]. Let us now formally derive the expression for the electron density of the beam at time t as a function of the longitudinal coordinate z. Taking a conservation of charge into account, one has
d[kz(z0 , t)] −1 , n(z0 , t) = n0 (z0 ) (9) d(kz0 ) where n0 (z0 ) is the initial density of the beam at point z0 . In our case, the longitudinal coordinate z is a function of θ so one has from Eqs. (2) d[kz(z0 , t)] ∂[kZ(θ)] ∂(ωt) −1 =1− · d(kz0 ) ∂θ ∂θ −1 pz (θ ) =1− (10) = γ (θ ) 1 + pz (θ ) and now θ is the function of t and kz0 . Then from Eqs. (9) and (10) one has n(z0 , t) = n0 (z0 )γ (θ ),
(11)
where θ is defined by the last equation of Eqs. (2). To derive the density distribution at time t, depending on coordinates z, one needs to switch from parameter z0 to the independent parameter z {lying, of course, inside the allowed interval [zmin (t), zmax (t)]}. Then one has n(z, t) = n0 z0 (z) γ (ωt − kz), z ∈ zmin (t), zmax (t) . (12) This expression has the simplest form when the initial density is constant inside some interval z0 ∈ [0, zmax ] and zero outside. Then the density distribution is n(z, t)/n0 = γ (ωt − kz), z ∈ zmin (t), zmax (t) , (13) and the ends of the interval are just the coordinates of the left and the right electrons at time t [and can be derived from Eqs. (2) along with the function γ ]. From Eq. (13) it is obvious that, in the electron beam, there is a density wave, which is running inside the laser pulse with the same speed. Of course, each electron in this density wave has the speed less than c, and slips along the density ripple alternately entering the regions with small and large density. The energy and longitudinal momentum pz = γ − 1 of an electron are maximal where the density is high and minimal in the regions with small density. The form of the density modulation is defined solely by the transverse momentum py [cf. Eq. (2)]. So if py is without a constant term, then the period of density modulation is λ/2. Otherwise there will be two spatial harmonics with periods λ/2 and λ, and relative amplitudes of these harmonics will be defined by the parameters of the laser pulse front. Besides, by changing the form of the laser pulse envelope, one can control the average density of the electron beam (produce slow density modulation). For a constant amplitude of the
laser pulse, the average density will be constant (the Coulomb forces will give a slow density modulation of the beam even in the constant amplitude field, however, in our analysis, this effect is omitted because a small enough electron density of the beam is considered). In passing, it is interesting to note that the measurement of the amplitudes of density modulation harmonics by, e.g., detecting the coherent transition radiation of the beam [23,26,27] can give very valuable information about the parameters of the laser pulse front. 4. Density distributions for electron beams: Theory versus PIC simulations Let us at first consider the formation of a single short electron bunch with the length smaller than the laser wavelength. To realize this, the initial length Lin of the electron beam must be smaller than the maximal length L2π defined for the laser pulse. Then, the density distribution will change its form with the slow time scale T2π . For Lin λ, the compressed length of the beam will oscillate between Lin and Lin /γ and the density will oscillate between n0 and n0 γ . For Lin ∼ L2π , the density distribution will have the width about λ, and will have one or two maxima. It is necessary to note that to output the electron beam from the laser pulse, one needs to place a plasma separator with an accuracy much better than L2π (that seems not to be a very difficult problem). Below, the electron density distributions inside the laser pulse with the amplitude a0 = 5 are presented for different times. In Fig. 1, the laser pulse has the step-like envelope; in Fig. 2, the laser front has a Gaussian shape and the duration 5λ (at half maximum). The values L2π for both cases can be defined from Eqs. (2) and (6). These values are about 19λ and 6λ. In Fig. 1, the initial length of the beam is 0.2λ, and it is 3λ in Fig. 2. In the same figures, the results of the PIC simulations with the one-dimensional version of XOOPIC code [46] are also presented (solid cyan lines). The densities of the electron beam were n0 = 1015 cm−3 for Fig. 1 and n0 = 1012 cm−3 for Fig. 2, initial velocity of the electrons is equal to zero. There were 8100 (Fig. 1) or 900 (Fig. 2) spatial grid points on the laser wavelength λ, and there were 10 (Fig. 1) or 20 (Fig. 2) macroparticles in one grid cell. From these figures, one can conclude that the theory gives a very good agreement with the PIC simulations. The excess noise in Fig. 1(d) and (e) can be explained by not high enough accuracy of the PIC simulations (to remove the noise it is necessary to increase the number of spatial grid points on the laser wavelength). From Fig. 1, one can conclude that the electron beam is almost monoenergetic at the point of maximum compression [Fig. 1(c)]. More exactly, the spread in the energy is smaller for the smaller initial electron beam length and the larger amplitude of the laser pulse. Besides, at the points of intermediate compression, there is a modulation in the energy of the electrons [cf. Fig. 1(b) and (d), the energy is equal to the density according to Eq. (13)]. This energy modulation allows, in principle, to realize a further compression with the help of a magnetic chi-
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Fig. 1. Pulsation of the short electron beam in the field of the ultra-intense laser pulse with a step-like envelope and a0 = 5 (blue dots—theory, solid cyan lines—PIC simulations). Electron density (in units of n0 ) is presented for different times: ωt 1.26, 6.28, 62.83, 116.24, 124.41 for (a)–(e). Distance on the horizontal axis is measured from the head of the laser pulse, which is located in kz = 0. Note different vertical scales for plots (a)–(e). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
cane. In Fig. 2, the gradual compression of the electron beam by the laser front can be easily traced. Here also, the average density of the electron beam is modulated [Fig. 2(b) and (c)]. Formation of the microbunched electron beams is presented in Fig. 3. In Fig. 3(a), the envelope of the laser pulse is a steplike; in Fig. 3(b), the duration of the laser pulse front is 0.89λ (at half maximum), and in Fig. 3(c) it is 2.2λ. The initial length Lin of the electron beam is equal to 100λ (0.1 mm for λ = 1 µ). The results of the PIC simulations are presented for the case of Fig. 3(c) (solid cyan line). The electron density for this case is n0 = 1010 cm−3 . There are 510 spatial grid points on the laser wavelength λ, and there are 20 macroparticles in one grid cell. Again, the results of the theory and the PIC simulation are almost the same. From these figures, the microbunching of the electron beams is evident. Besides, for the longitudinal modulation of the electron density, the relative amplitudes of two spatial harmonics with periods λ and λ/2 (frequencies ω and 2ω) depend on the duration of the laser pulse front as predicted by the theory.
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Fig. 2. Compression of the electron beam by the laser pulse front with duration 5λ (at half maximum) and a0 = 5 (blue dots—theory, solid cyan lines—PIC simulations). The laser field at ωt = 188.5 is presented in (a); (b)–(g)—electron density distributions (in units of n0 ) for different times ωt 119.41, 128.84, 150.83, 216.8, 270.21, 282.78. Note different vertical scales for plots (b)–(g). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Again, one can see that by changing the laser pulse envelope, it is possible to control the average density of the electron beam [cf. slow density modulation at the head of the electron beam in Fig. 3(c)]. 5. Discussions of results and conclusions It was demonstrated above that the ultra-intense laser pulses can generate microbunched relativistic electron beams or ultrashort electron bunches. Let us discuss now the conditions, which are necessary for practical realization of this method, and possible parameters of resulting electron beams. The required laser power is determined by the amplitude a0 and the diameter of the laser beam d, which has to be at least not smaller than the
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Fig. 3. Microbunching of the electron beam by laser pulses with different front duration: (a)—the step-like envelope, (b) and (c)—the duration of the laser pulse front at half maximum is 0.89λ and 2.2λ. The electron densities (in units of n0 ) are presented for ωt 100 · 2π . Note different vertical scales in plots (a)–(c).
transverse displacement y of the electrons in the beam. Therefore from Eqs. (2), one can conclude that the required value of d depends on the duration of the laser pulse front and also on the initial length of the electron beam (through maximal value of θ ). One can estimate from Eqs. (5) that for the generation of the ultrashort electron bunch by the laser pulse with a step-like envelope, it is necessary to have d a0 , where d is measured in λ, and θ = 2π is used. Then P = (da0 )2 P1 = a04 P1 , where P1 is the laser power required to create a0 = 1 in the beam spot with diameter λ (about 14 GW for λ 1 µ). So, for a 1 PW laser pulse, which is available now, the maximum value of a0 can be about 17 and d 17 µ, giving the maximal electron energy on the order of 300 MeV. If the laser power is 1 EW (1018 W) [38] then a0 100 (d 0.1 mm) and the electron energy is 10 GeV. For the generation of a microbunched electron beam by a laser pulse with a step-like envelope, the requirements on d are more strict, namely d must be proportional to the maximal value of θ [cf. Eqs. (5)]. Thus, for the generation of the beam with, e.g., 5 humps one needs to have θmax = 5 · 2π , and d 5a0 . So the maximal value of a0 for the petawatt laser is only a0 8 (d 40 µ). For laser pulses with a smooth front,
the constant term in the transverse momentum is vanishing, so the transverse coordinate of an electron is bounded, and the required value of d is independent on the length of the beam [cf. Eqs. (2)]. So all estimates for θ = 2π (more exactly, for θ = 1) and a step-like envelope are valid also for this case. The longitudinal scale of the method is defined by the value of L2π . Again, for the ultrashort electron bunch generated by the laser pulse with a step-like envelope, L2π is about 225 µ for 1 PW laser pulse and 7.5 mm for 1 EW laser pulse. If the required number of humps in the beam is greater than 1 then this length is increased proportionally. It is necessary to note that the generalization of the above results on the case of nonzero initial velocity of the electrons is straightforward. Also, all the results can be easily expanded on the case of short laser pulses. In this case, noticeable increase of the electron density will be only in the regions where the amplitude of the laser pulse is large enough. For this region, all the results for microbunching are applicable. When the laser pulse will go out of the electron beam, the electron density will return to its initial value. In conclusion, the formation of the coherent structures, induced by a super-intense plane electromagnetic wave with a sharp rising edge in an ensemble of electrons (electron beam) in vacuum, was considered. The theory describing this process was elaborated. It was shown that the laser pulse can strongly compress the electron beam, and this process takes place just at the front of the laser pulse. If the laser pulse has a slowly varying envelope then the density of the compressed electron beam has a slow modulation, because the degree of compression and the average density of the electron beam depend on the laser pulse amplitude. Also, an ultraintense laser pulse generates fast density modulation (microbunching) in the electron beam. Depending on the duration of a laser pulse front, two spatial harmonics can be present simultaneously in the longitudinal density modulations of the electron beam—one with laser wavelength and the other with half of the laser wavelength. Besides, by varying the laser pulse amplitude and the initial length of the electron beam, it is possible to change the number of microbunches in the compressed electron beam, so that for certain conditions only one electron microbunch can be produced with an ultrashort length on the attosecond level. For some moments, the energy of the electrons in this single bunch is strongly modulated, allowing further compression of the bunch with a magnetic chicane. The results of the theory were compared with the 1D PIC simulations and a good agreement was found. Acknowledgements This work was partially supported by the CRI Program of the Korean Ministry of Science and Technology, by the INTAS project 01-0233 (V.A.C.), and by RFBR project 05-02-17297-a. References [1] L.D. Landau, E.M. Lifshitz, Teoriya Polya, Nauka, Moscow, 1988, The Classical Theory of Fields, Pergamon, Oxford, 1975.
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